let Y be non empty set ; :: thesis: for a being Element of Funcs (Y,BOOLEAN) holds a '&' ('not' a) = O_el Y
let a be Element of Funcs (Y,BOOLEAN); :: thesis: a '&' ('not' a) = O_el Y
consider k3 being Function such that
A1: a '&' ('not' a) = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: O_el Y = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for x being Element of Y holds (a '&' ('not' a)) . x = (O_el Y) . x

proof

let x be Element of Y; :: thesis: (a '&' ('not' a)) . x = (O_el Y) . x
A5: (a '&' ('not' a)) . x = (a . x) '&' (('not' a) . x) by MARGREL1:def 20
.= (a . x) '&' ('not' (a . x)) by MARGREL1:def 19 ;

now

per cases ( a . x = TRUE or a . x = FALSE ) by XBOOLEAN:def 3;

case a . x = TRUE ; :: thesis: (a '&' ('not' a)) . x = (O_el Y) . x

hence (a '&' ('not' a)) . x = (O_el Y) . x by A5, BVFUNC_1:def 10; :: thesis: verum

end;

case a . x = FALSE ; :: thesis: (a '&' ('not' a)) . x = (O_el Y) . x

hence (a '&' ('not' a)) . x = (O_el Y) . x by A5, BVFUNC_1:def 10; :: thesis: verum

end;

end;

end;

hence (a '&' ('not' a)) . x = (O_el Y) . x ; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence a '&' ('not' a) = O_el Y by A1, A2, A3, A4, FUNCT_1:2; :: thesis: verum